Galileo’s discovery of scaling laws

Galileo Galilei, Two New Sciences, translated by Henry Crew and Alfonso de Salvio (Dover, New York, 1954). Originally published by Elzevir, 1638.

Reference 1, pp. xx–xxi.

Reference 1, p. xii. The full title is “Discourses and Mathematical Demonstrations concerning Two New Sciences pertaining to Mechanics and Local Motions.” Galileo complained that the publishers had substituted “a low and common title for the noble and dignified one carried upon the title-page.” His preferred title is unfortunately lost.

Galileo Galilei, “Due lezioni all’Accademia Fiorentina circa la figura, sito e grandezza dell’Inferno di Dante,” in Le Opere di Galileo Galilei, edited by G. Barbèra (Ristampa della Edizione Nazionale, Florence, 1933), Vol. 9, pp. 31–57. Translated by Mark A. Peterson, http://www.mtholyoke.edu/ ̃mpeterso/classes/galileo/inferno.html.

Reference 1, p. 90.

Reference 1, p. 2.

Reference 1, p. 112.

Reference 1, p. 3.

Archimedes’ law of the lever, the balance of torques, was second nature to Galileo.

We would now say the stress itself varies linearly across the beam, so that the correct dependence is the second moment of area, going as the diameter to the fourth power.

All propositions in this paragraph are from the Second Day.

Reference 1, p. 6.

Aristotle, in Minor Works, translated by W. S. Hett (Harvard U.P., Cambridge, MA, 1936), pp. 330–411.

Reference 1, p. 135.

Reference 1, p. 5.

Reference 1, p. 7.

Galileo Galilei, in Discoveries and Opinions of Galileo, translated by Stillman Drake (Anchor Books, New York, 1957).

Galileo Galilei, Discourse on Bodies in Water, translated by Thomas Salusbury (University of Illinois Press, Urbana, IL, 1960).

Galileo Galilei, Dialogue Concerning the Two Chief World Systems, translated by Stillman Drake (University of California Press, Berkeley, CA, 1967).

M. A. Finocchiaro, The Galileo Affair (University of California Press, Berkeley, CA, 1989).

“Le Opere di Galileo Galilei,” edited by G. Barbèra (Ristampa della Edizione Nazionale, Florence, 1933), Vol. 10, No. 207, pp. 228–230.

Stillman Drake, Galileo at Work (Dover, Mineola, NY, 1978), pp. 16–17.

Reference 1, p. 100.

Reference 16, p. 13.

Mario Biagioli, Galileo Courtier (University of Chicago Press, Chicago, 1993), pp. 30–31.

Reference 16, pp. 106–107.

In fact Manetti was a Florentine intellectual of the previous century, the biographer of Filippo Brunelleschi. He lived well before the founding of the Florentine Academy.

Vellutello was from Lucca, a notable rival of Florence.

Reference 1, p. 3.

Reference 19.

See, for example, the entertaining account in William Dunham, Journey Through Genius (Wiley, New York, 1990).

The way he uses scaling goes as follows. Galileo’s opponents argue that the lamina floats because it is broad and flat, and the water resists being cut. But where the water is cut is along the perimeter, so it is supported at the perimeter, and thus, by scaling, the lamina floats better if it is divided into smaller pieces, that is, it floats better if it is less broad and flat, contradicting their own position.

Reference 1, p. 125.

“Le Opere di Galileo Galilei,” edited by G. Barbèra (Ristampa della Edizione Nazionale, Florence, 1933), Vol. 9, p. 7.

Because he could not obtain the text, Alamanni provided his correspondent a summary of the lectures (five years after they were delivered!). “It consisted in this, that he reviewed the opinion of the Florentine Antonio Manetti concerning the site of Dante’s Inferno, published in a book printed by the Giunti, and then he reviewed the opinion of Vellutello, a commentator on Dante, on the same subject, and comparing the one with the other, he showed that of Manetti to be better.”

Reference 1, pp. 130–131.

See for example the exchange between Simplicio and Sagredo, Ref. 1, bottom of p. 137.